Non-Hermitian Bosonic Kitaev Model

• The non-Hermitian bosonic Kitaev model is a bosonic analogue of the fermionic chain, where intrinsic non-Hermiticity arises from bosonic pairing and commutation rules.
• It exhibits phenomena such as the non-Hermitian skin effect, phase-dependent chiral transport, and exceptional points that signal transitions in localization and topology.
• Experimental implementations in superconducting, optomechanical, and photonic systems validate its potential for probing novel non-Hermitian topological and amplification effects.

The non-Hermitian bosonic Kitaev model denotes a class of one-dimensional bosonic lattices with hopping and pairing terms that are constructed as bosonic analogues of the fermionic Kitaev chain, but whose natural Bogoliubov–de Gennes, dynamical, or excitation matrices are generically non-Hermitian even when the underlying many-body Hamiltonian is Hermitian. In this setting, non-Hermiticity is not restricted to explicit gain, loss, or asymmetric couplings: it also emerges intrinsically from bosonic commutation relations, paraunitary Bogoliubov structure, and number-nonconserving quadratic terms. The resulting models exhibit non-Hermitian skin effect, phase-dependent chiral transport, exceptional points, localization–delocalization transitions in Fock space, and analytically tractable topological edge phenomena, with direct implementations in superconducting, optomechanical, photonic, and cold-atom platforms (McDonald et al., 2018, Yokomizo et al., 2020, Busnaina et al., 2023).

1. Canonical model and principal generalizations

A standard bosonic Kitaev chain is a quadratic bosonic Hamiltonian with nearest-neighbor hopping and pairing. In one widely used gauge, the Hamiltonian is

$$\hat{\mathcal{H}}=\sum_j \left( i t\, \hat{a}_{j+1}^\dagger \hat{a}_j + i\Delta \,\hat{a}_{j+1}^\dagger \hat{a}_j^\dagger + \text{h.c.} \right),$$

with bosonic operators $\hat a_j,\hat a_j^\dagger$, hopping amplitude $t$, and pairing amplitude $\Delta$. A generalized form adds an onsite bosonic frequency or chemical-potential-like term, for example $\omega \sum_j a_j^\dagger a_j$ or $\mu \sum_j (2 b_j^\dagger b_j+1)$ (Fortin et al., 2024, He et al., 2024).

The bosonic analogue differs from the fermionic Kitaev chain in two structural ways that recur throughout the literature. First, bosons allow unbounded occupation, so pairing can produce dynamical instability unless parameters are restricted; a representative stable regime is $0<\Delta<t$ for the nearest-neighbor chain (Fortin et al., 2024). Second, bosons are diagonalized by bosonic Bogoliubov transformations rather than ordinary unitary fermionic Bogoliubov transformations, and this is the origin of the effective non-Hermitian structure (Wang et al., 2022).

Several extensions define the current research landscape. A staggered bosonic Kitaev chain introduces dimerized hopping, pairing, and onsite potentials,

$\alpha_j=b_{2j-1},\qquad \beta_j=b_{2j},$

with parameters $t_1,t_2,\Delta_1,\Delta_2,g_1,g_2$ alternating between odd and even sublattices. In the formulation studied in 2025, the kinetic sector contains

$$i t_1 b_{2j-1}^\dagger b_{2j}+i t_2 b_{2j}^\dagger b_{2j+1} +i\Delta_1 b_{2j-1}^\dagger b_{2j}^\dagger+i\Delta_2 b_{2j}^\dagger b_{2j+1}^\dagger+\mathrm{H.c.},$$

while the potential sector carries staggered onsite terms $\hat a_j,\hat a_j^\dagger$0 and $\hat a_j,\hat a_j^\dagger$1 (Wang et al., 11 May 2025).

A second important generalization is the modified bosonic Kitaev–SSH chain with two sublattices $\hat a_j,\hat a_j^\dagger$2 per unit cell,

$\hat a_j,\hat a_j^\dagger$3

which supports both nonHermitian skin effect and nontrivial topological edge modes in its excitation Hamiltonian (Bomantara et al., 21 May 2025).

Periodically driven variants form a further branch of the subject. A binary-driven modified bosonic Kitaev chain yields a non-unitary Floquet excitation operator even though the microscopic Floquet operator is unitary, and supports nonHermitian Floquet topological phenomena including zero modes, $\hat a_j,\hat a_j^\dagger$4 modes, and skin effect (Bomantara, 19 Nov 2025).

2. Bosonic BdG structure and the origin of effective non-Hermiticity

The defining theoretical fact is that, for bosons, the matrix governing excitations is not the Hermitian BdG matrix itself but a non-Hermitian dynamical or excitation matrix. In a general quadratic bosonic Hamiltonian,

$\hat a_j,\hat a_j^\dagger$5

the bosonic BdG matrix obeys

$\hat a_j,\hat a_j^\dagger$6

but the eigenvalue problem relevant for quasiparticles is that of $\hat a_j,\hat a_j^\dagger$7, or equivalently $\hat a_j,\hat a_j^\dagger$8 in momentum space, and this matrix is generically non-Hermitian despite Hermiticity of the many-body Hamiltonian (Yokomizo et al., 2020).

This intrinsic non-Hermiticity can be stated at the operator level. For linear combinations of quadratures or Nambu operators, the commutator eigenvalue problem

$\hat a_j,\hat a_j^\dagger$9

defines excitation energies. In the modified bosonic Kitaev chain, this becomes a non-Hermitian matrix eigenproblem $t$0, where $t$1 is the excitation Hamiltonian. The same logic underlies the Floquet generalization

$t$2

with $t$3 generically non-unitary although the microscopic Floquet operator is unitary (Bomantara et al., 21 May 2025, Bomantara, 19 Nov 2025).

In quadrature language, the bosonic Kitaev chain decouples into two nonreciprocal chains. With

$t$4

the nearest-neighbor model obeys

$t$5

Thus the $t$6-chain has nonreciprocal hopping in one direction and the $t$7-chain in the opposite direction. This is the simplest direct manifestation of effective non-Hermiticity in a Hermitian bosonic model (Fortin et al., 2024).

The general stability theory is formulated in terms of pseudo-Hermiticity and generalized $t$8 symmetry. For quadratic bosonic Hamiltonians, the effective BdG generator $t$9 satisfies

$\Delta$0

so it is pseudo-Hermitian with respect to the indefinite metric $\Delta$1. Dynamical stability means that $\Delta$2 is diagonalizable and all its eigenvalues are real; dynamical instability means that $\Delta$3 has at least one complex eigenvalue or a non-trivial Jordan chain. Within this framework, stability-to-instability transitions occur both at exceptional points and at Krein collisions, and the paper introducing Krein phase rigidity shows that boundary conditions supporting Majorana zero modes in the fermionic Kitaev chain are precisely the same that support stability in the bosonic chain (Flynn et al., 2020).

A common misconception is that “non-Hermitian bosonic Kitaev model” must refer to a non-Hermitian many-body Hamiltonian. In a large part of the literature, it instead refers to Hermitian bosonic pairing Hamiltonians whose excitation, dynamical, or core matrices are non-Hermitian. This distinction is explicit in both the static and Floquet constructions (Fortin et al., 2024, Bomantara, 19 Nov 2025).

3. Non-Bloch topology, skin effect, and edge physics

The most conspicuous consequence of effective non-Hermiticity is the non-Hermitian skin effect. In the bosonic Kitaev chain, all eigenmodes can be exponentially localized to the edges of an open chain even without dissipation. This happens because the decoupled quadrature chains are governed by non-Hermitian dynamical matrices with asymmetric hopping, so periodic- and open-boundary spectra differ strongly (Fortin et al., 2024).

The original bosonic Kitaev–Majorana construction also emphasized phase-dependent chirality. In momentum space the dynamical matrix can take the form

$\Delta$4

with eigenvalues

$\Delta$5

The corresponding real-space dynamics exhibits phase-dependent chiral transport and a drastic sensitivity to boundary conditions: a boundary-less system may have only delocalized, dynamically unstable modes, while a finite open chain is described by localized, dynamically stable modes (McDonald et al., 2018).

The correct bulk theory is therefore non-Bloch rather than ordinary Bloch band theory. For bosonic BdG systems, the generalized Brillouin zone is defined by the condition

$\Delta$6

where the $\Delta$7 are the roots of the non-Bloch characteristic equation ordered by modulus. The energy winding number

$\Delta$8

diagnoses spectral winding and predicts when open-boundary spectra differ from periodic-boundary spectra. Applied to the bosonic Kitaev–Majorana chain, this formalism reveals instability against infinitesimal perturbations and reentrant behavior of the non-Hermitian skin effect (Yokomizo et al., 2020).

Topological structure becomes analytically sharper in the modified bosonic Kitaev chain. There, the excitation Hamiltonian is mapped exactly to a nonHermitian SSH model through non-unitary similarity transformations. In the fully general $\Delta$9 case, the transformed problem is

$\omega \sum_j a_j^\dagger a_j$0

with

$\omega \sum_j a_j^\dagger a_j$1

The topological phase boundary is

$\omega \sum_j a_j^\dagger a_j$2

and the winding numbers are nontrivial when $\omega \sum_j a_j^\dagger a_j$3. In the original, non-Hermitian frame, the same system displays skin effect because the similarity transformation is site-dependent and non-unitary (Bomantara et al., 21 May 2025).

The regular bosonic Kitaev chain and its dissipative extension are also classified directly at the level of the dynamical matrix. In that setting the relevant non-Hermitian symmetry class is DIII$\omega \sum_j a_j^\dagger a_j$4, with a 1D $\omega \sum_j a_j^\dagger a_j$5 invariant given by a winding number of complex energies. This is a topological invariant of the dynamical matrix $\omega \sum_j a_j^\dagger a_j$6, not of the microscopic Hermitian Hamiltonian, and it underlies topological amplification and the non-Hermitian skin effect without dissipation (Fortin et al., 2024).

A second common misconception is to identify bosonic edge physics with fermionic Majorana zero modes without qualification. The literature instead emphasizes that bosons fractionalize into quadrature operators and that the relevant protected or amplified boundary structures live in the excitation or dynamical problem. This suggests that “Majorana-like” language in the bosonic case is heuristic unless it is explicitly tied to quadrature or BdG operators (McDonald et al., 2018, Slim et al., 2023).

4. Exceptional points and many-body criticality

Exceptional points provide a second organizing principle. In the Hermitian bosonic Kitaev chain with nearest-neighbor hopping and pairing,

$\omega \sum_j a_j^\dagger a_j$7

Fourier transformation yields a $\omega \sum_j a_j^\dagger a_j$8 non-Hermitian core matrix

$\omega \sum_j a_j^\dagger a_j$9

Its eigenvalues are

$\mu \sum_j (2 b_j^\dagger b_j+1)$0

and the exceptional point appears when

$\mu \sum_j (2 b_j^\dagger b_j+1)$1

At that point the core becomes non-diagonalizable and the hidden exceptional point coincides with a localization–delocalization transition in an equivalent single-particle problem built in a BCS-like pairing basis (He et al., 2024).

The pairing-basis mapping is central because it converts each momentum sector into a semi-infinite tight-binding chain in Fock space. In one regime the equivalent Hamiltonian is

$\mu \sum_j (2 b_j^\dagger b_j+1)$2

whose eigenstates are localized in the pair-number coordinate $\mu \sum_j (2 b_j^\dagger b_j+1)$3; beyond the exceptional point, the corresponding eigenstates become extended. The inverse participation ratio and mean inverse participation ratio then furnish an explicit localization diagnostic for the hidden EP-driven quantum phase transition (He et al., 2024).

The two-site bosonic Kitaev dimer gives a minimal version of the same mechanism. For

$\mu \sum_j (2 b_j^\dagger b_j+1)$4

a Nambu transformation produces two non-Hermitian $\mu \sum_j (2 b_j^\dagger b_j+1)$5 core matrices,

$\mu \sum_j (2 b_j^\dagger b_j+1)$6

Their exceptional points are

$\mu \sum_j (2 b_j^\dagger b_j+1)$7

These EPs divide parameter space into four regions in which the equivalent Hamiltonians are combinations of a harmonic oscillator and an inverted harmonic oscillator, and the second-order intensity correlation $\mu \sum_j (2 b_j^\dagger b_j+1)$8 provides a dynamical witness of the EPs (He et al., 26 Feb 2025).

The most comprehensive many-body formulation to date appears in the staggered bosonic Kitaev chain. There the non-Hermitian object is a $\mu \sum_j (2 b_j^\dagger b_j+1)$9 Bloch core matrix defined in the Nambu framework. The analysis derives explicit analytic conditions for the emergence of exceptional points, with each EP marking the onset of complex-conjugate eigenvalue pairs. By mapping the full many-body Hamiltonian onto an effective tight-binding network in Fock-space and introducing layer-resolved inverse participation ratio, the work demonstrates that these EPs coincide precisely with sharp localization–delocalization transitions of collective eigenstates, and that numerical analyses across hopping amplitudes, pairing strengths, and onsite potentials confirm that the EP of effective Hamiltonian universally capture the global many-body phase boundaries (Wang et al., 11 May 2025).

These results establish a recurring pattern: in bosonic Kitaev systems, a Hermitian many-body Hamiltonian can inherit non-analytic transitions from a non-Hermitian core. This suggests a broader exceptional-point criterion for quantum criticality in interacting bosonic lattices, provided the relevant BdG or excitation matrix remains analytically accessible (Wang et al., 11 May 2025).

5. Stability, interactions, dissipation, disorder, and driven extensions

Interactions alter the non-Hermitian phenomenology in a way that is specific to bosons. In bosonic Kitaev-Hubbard models on a chain and a two-leg ladder, the hard-core limit maps exactly to spin-$0<\Delta<t$0 models and the non-Hermitian skin effect disappears. The same study shows that hard-core bosons can engineer the Kitaev interaction, the Dzyaloshinskii-Moriya interaction and the compass interaction in the presence of the complex hopping and pairing terms, while the quantum criticalities of the chain with a three-body constraint and unconstrained soft-core bosons were investigated by the density matrix renormalization group method (Wang et al., 2022).

Dissipation introduces a different control parameter. For the bosonic Kitaev chain with on-site Markovian loss, the dynamical matrix is

$0<\Delta<t$1

and the response depends strongly on the dissipation pattern. Uniform and non-uniform dissipation can either preserve or destroy topological amplification. When the dissipation is placed on every other site, the system remains topological even for very large dissipation which exceeds the system's non-Hermitian gap and the exponential amplification persists. By contrast, dividing the chain into unit cells of an odd number of sites and placing dissipation on the first site leads to a topological phase transition at some critical value of the dissipation (Fortin et al., 2024).

Floquet driving extends the same logic to periodically driven bosonic chains. In the driven modified bosonic Kitaev chain, the excitation-level Floquet operator is non-unitary and supports nonHermitian Floquet topological phenomena. One case study displays coexistence of nonHermitian skin effect, topological zero modes, and topological $0<\Delta<t$2 modes; another supports multiple topological zero modes and topological $0<\Delta<t$3 modes in a tunable manner. Onsite bosonic frequency and spatial disorder affect these structures differently: in one model the skin effect is easily suppressed and revived by, respectively, onsite frequency and spatial disorder, while in the other it can be insensitive to both perturbations (Bomantara, 19 Nov 2025).

Disorder also plays a nontrivial role in static modified chains. For the modified bosonic Kitaev chain at $0<\Delta<t$4, moderate disorder preserves topological zero modes as long as the SSH-like gap near $0<\Delta<t$5 stays open. At nonzero $0<\Delta<t$6, disorder can partially recover NHSE-like localization even though the clean onsite potential destroys the skin effect. This is one of the clearest examples in which disorder does not simply compete with non-Hermitian topology but can reconstruct part of it (Bomantara et al., 21 May 2025).

A further extension interprets the $0<\Delta<t$7 quadrature equations as two Hatano–Nelson chains and then as projections of Schrödinger equations on hyperbolic surfaces. In that formulation a finite chemical potential couples two hyperbolic surfaces, and the resulting quantum sensor has sensitivity that grows exponentially with system size. A plausible implication is that some “non-Hermitian bosonic Kitaev” effects can be reformulated geometrically rather than solely spectrally (Lv et al., 2024).

6. Experimental realizations and conceptual status

The bosonic Kitaev chain has been realized experimentally in at least two complementary architectures. In a multimode superconducting parametric cavity, lattice sites are mapped to frequency modes, and tunable complex hopping and pairing are created by parametric pumping at mode-difference and mode-sum frequencies. The experiment demonstrated chiral transport, quadrature wavefunction localization, and sensitivity to boundary conditions, and framed these observations as precursors of nontrivial topology and the non-Hermitian skin effect in a fully Hermitian bosonic Hamiltonian (Busnaina et al., 2023).

A nano-optomechanical network provides a second realization. There the bosonic Kitaev–Majorana chain

$0<\Delta<t$8

was implemented using beamsplitter and two-mode squeezing interactions. The experiment observed quadrature-dependent chiral amplification, exponential scaling of the gain with system size, strong sensitivity to boundary conditions, a dynamical phase diagram linked to non-Hermitian topological phase transitions, and an exponentially enhanced response to a small perturbation (Slim et al., 2023).

The hidden-EP program also admits a realistic detection route through the Dicke model. After Holstein–Primakoff reduction, the effective two-mode Hamiltonian

$0<\Delta<t$9

inherits two non-Hermitian $\alpha_j=b_{2j-1},\qquad \beta_j=b_{2j},$0 core matrices with a transition at $\alpha_j=b_{2j-1},\qquad \beta_j=b_{2j},$1. Numerical simulations then show that the average number of photons after a quench from the empty state exhibits a clear transition point at the EP (He et al., 2024).

Within the present literature, the conceptual status of the subject is therefore twofold. On one side are explicitly non-Hermitian or dissipative bosonic Kitaev-type models. On the other are Hermitian bosonic Hamiltonians whose dynamical matrix, excitation Hamiltonian, Floquet excitation operator, or Nambu core matrix is non-Hermitian. The latter class is not a marginal reinterpretation; it is the route by which non-Hermitian skin effect, exceptional points, and topological amplification arise in most bosonic Kitaev realizations discussed to date (Bomantara et al., 21 May 2025, Fortin et al., 2024).

For that reason, the term “non-Hermitian bosonic Kitaev model” is best understood as referring to a research program rather than a single Hamiltonian. Its unifying content is a bosonic Kitaev-type pairing structure together with a non-Hermitian effective spectral problem, from which follow boundary-sensitive spectra, quadrature-selective transport, exceptional-point criticality, and experimentally measurable amplification and localization phenomena (McDonald et al., 2018, Wang et al., 11 May 2025).